L(s) = 1 | + (1.97 + 1.14i)2-s + (1.57 + 2.72i)3-s + (1.61 + 2.78i)4-s + (−1.84 − 1.06i)5-s + 7.19i·6-s + 2.78i·8-s + (−3.46 + 5.99i)9-s + (−2.42 − 4.20i)10-s + (0.267 − 0.154i)11-s + (−5.07 + 8.78i)12-s + (3.22 − 1.62i)13-s − 6.69i·15-s + (0.0349 − 0.0605i)16-s + (0.887 + 1.53i)17-s + (−13.7 + 7.91i)18-s + (−1.54 − 0.890i)19-s + ⋯ |
L(s) = 1 | + (1.39 + 0.807i)2-s + (0.909 + 1.57i)3-s + (0.805 + 1.39i)4-s + (−0.823 − 0.475i)5-s + 2.93i·6-s + 0.985i·8-s + (−1.15 + 1.99i)9-s + (−0.767 − 1.32i)10-s + (0.0805 − 0.0465i)11-s + (−1.46 + 2.53i)12-s + (0.893 − 0.449i)13-s − 1.72i·15-s + (0.00874 − 0.0151i)16-s + (0.215 + 0.372i)17-s + (−3.22 + 1.86i)18-s + (−0.353 − 0.204i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26053 + 3.41347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26053 + 3.41347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-3.22 + 1.62i)T \) |
good | 2 | \( 1 + (-1.97 - 1.14i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.57 - 2.72i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.84 + 1.06i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.267 + 0.154i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.887 - 1.53i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.54 + 0.890i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.575 + 0.996i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 + (3.98 - 2.30i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.79 - 2.77i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.72iT - 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + (-8.24 - 4.75i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.72 + 6.44i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.03 + 4.06i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.72 - 2.97i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.9 - 6.30i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.35iT - 71T^{2} \) |
| 73 | \( 1 + (10.2 - 5.94i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.96 + 6.86i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + (1.43 + 0.829i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.66iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91691964839685994903530652116, −10.08154551315492211065225439515, −8.880363964006714685820575508386, −8.324755659027042710549498332871, −7.46006420872393361351492780071, −6.05817102724019930261021639668, −5.13332141068775544941280646659, −4.27862682983416718414472555427, −3.80201704417294302340007405973, −2.90442222569716802167963161778,
1.38036835468886169391786320184, 2.51679429682294839959998485319, 3.37993960297112596608116607078, 4.15982408447007576050090090078, 5.77535379640052183216096428057, 6.60494610516343778272963534680, 7.47774184648565643762617127837, 8.261879839956605550698509279765, 9.283490762263529636745308300973, 10.79839116777049109611860553373